##  [Heat Kernel](/heat-kernel) 

  ##  [Heat Kernel](https://natural.quantumdictionary.io/heat-kernel-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The fundamental solution (integral kernel) of the heat (diffusion) equation for a given elliptic operator; it gives the temperature (or density) at a point and time produced by an instantaneous unit source at another point.

 

 

 

 

 





 

 



 ##  [Heat Kernel](https://puremath.quantumdictionary.io/heat-kernel-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The fundamental solution (integral kernel) of the heat equation that propagates initial data forward in time; equivalently the kernel of the heat semigroup exp(tL) for a Laplace-type operator L, providing time-dependent smoothing and short-time geometric asymptotics.